Thursday, June 5, 2008

Ch. 17 Ellipse - Concept Review

1. Introduction

2. Equation of ellipse in its standard form

x²/a² + y²/b² = 1

In x²/a² + y²/b² = 1, if a>b or a²>b² (denominator of x² is greater than that of y²), then the major and minor axes lie along x-axis and y-axis respectively.

The centre of the ellipse will be at (0,0).

Coordinates of the vertices will be at (a,0) and (-a,0).

Length of the major axis is 2a.
Length of the minor axis is 2b

Equation of major axis y = 0
Equation of minor axis x = 0

Eccentricity e = √(1 - b²/a²)

Length of latus rectum = 2b²/a

Equations of directrices x = a/e and x = -a/e

But if a

3. Second focus and second directrix of the ellipse

4. Vertices, major and minor axes, foci, directrices and centre of the ellipse

5. Ordinate, double ordinate and latus rectum of the ellipse

6. focal distances of a point on the ellipse

7. equation of ellipse in other forms

If the centre of the ellipse is at point (h,k) and the directions of axes are parallel to the coordinate axes, then its equation is

(x-h) ²/a² + (y-k) ²/b² = 1

8. Position of a point with respect to an ellipse

The point p(x1,y1) lies outside, on or inside the ellipse x²/a² + y²/b² = 1
According as x1²/a² + y1²/b²- 1>, = or ,0.

9.Parametric equations and parametric coordinates

The circle described on the major axis of an ellipse as diameter is called an auxiliary circle of the ellipse.

For the ellipse x²/a² + y²/b² = 1
Auxiliary circle is x² + y² = a²

Parametric coordinates of ellipse

If ф is the angle made by the line joining centre of ellipse with a point (x,y) on the ellipse x = a cos ф and y = b sin ф are the parametric coordinates of the ellipse x²/a² + y²/b² = 1, a>b

10. Equation of the chord joining any two points on an ellipse

If P(a cos θ, b sin θ) and Q( a cos ф,b sin ф) be any two points of the ellipse x²/a² + y²/b² = 1, then equation of chord joining P and Q is of the form

(y-y1) = m(x-x1) which is going to be

(y – b sin θ) = [(b sin ф - b sin θ)/( a cos ф - a cos θ)]*(x – a cos θ)

The equation can also be expressed as

(x/a)* cos [(θ+ ф)/2] +(y/b)*sin [(θ+ ф)/2] = cos [(θ- ф)/2]

11. Condition of a line to be a tangent to an ellipse

The condition for the line y = mx+c to be a tangent to the ellipse x²/a² + y²/b² = 1 is that c² = a²m²+b² or c = ±√( a²m²+b²)

12. Equation of tangent in terms of its slope

13. Equation of tangent at a point

14 Number of tangents drawn from a point to an ellipse

15. Equation of normal in different forms

16 Number of normals

17. Properties of eccentric angles of the conormal points

18. Equation of the pair of tangents from a point to an ellipse

19. Equation of the chord of contacts of tangents

19a. Equation of the chord bisected at a given point

20. Equation of diameter of an ellipse

21. Some properties of ellipse

No comments: